Origami & Folding
what one sheet can do
Generator

Box pleating

A generator in the designing a base library, called 20 times across 12 essays. Below: what it draws at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

box-pleating is one function. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and when the generator changes, this page changes with it.

At its defaults

Box pleatingDesigning on a grid, with every crease running along a grid line or at forty-five degrees to it. It gives up the efficiency of a free circle packing and gains something worth more for complex work — the creases meet where they are supposed to, and the errors do not accumulate.16 × 16 gridevery crease on a grid line, or at 45°which is why a 64-grid design can be folded at allmountainvalley

grid: 16

Box pleatingDesigning on a grid, with every crease running along a grid line or at forty-five degrees to it. It gives up the efficiency of a free circle packing and gains something worth more for complex work — the creases meet where they are supposed to, and the errors do not accumulate.16 × 16 gridevery crease on a grid line, or at 45°which is why a 64-grid design can be folded at allmountainvalley

grid: 32

Box pleatingDesigning on a grid, with every crease running along a grid line or at forty-five degrees to it. It gives up the efficiency of a free circle packing and gains something worth more for complex work — the creases meet where they are supposed to, and the errors do not accumulate.32 × 32 gridevery crease on a grid line, or at 45°which is why a 64-grid design can be folded at allmountainvalley

grid: 8

Box pleatingDesigning on a grid, with every crease running along a grid line or at forty-five degrees to it. It gives up the efficiency of a free circle packing and gains something worth more for complex work — the creases meet where they are supposed to, and the errors do not accumulate.8 × 8 gridevery crease on a grid line, or at 45°which is why a 64-grid design can be folded at allmountainvalley

What it checked while it drew

Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.

Where it is called

Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.

A grid glued

Box pleating is the designer's grid: every crease on a line, every angle a right angle or forty-five degrees, and a whole design method built on the convenience of it. Roll the grid into a tube and half the column counts stop folding, on a pattern whose whole selling point is that it always works.

Designing on a grid

Box pleating gives up the efficiency of a free packing and buys creases that land where they are supposed to. For a design with hundreds of folds that is not a compromise — it is the only thing that makes it foldable.

Ninety-nine in a hundred pass

A designer checks a box-pleated pattern the way every text teaches: vertex by vertex, counting mountains and valleys, watching the smallest sector. At sixteen divisions that check passes a hundred letterings in a hundred, and one of them folds. The check that separates them costs a single sweep over the crease list and is in no recipe anywhere.

Publishing the pattern instead of the sequence

A diagram sequence is one picture per step and a crease pattern is one picture. When designers began releasing patterns rather than diagrams, the cost of publishing a model fell by two orders of magnitude and the difficulty moved onto the reader — which is what made the complex era possible and what made most of it unfoldable.

Spelling a tree on a grid

Box pleating asks every limb of a design to be a whole number of grid squares, which sounds like rounding and is not. Rounding each limb to its own nearest whole number is one way to choose the numbers, and at most resolutions it is not the best way — the best whole-number version of a subject is often a coarser one, with fewer squares and a shape twice as close.

The crease has a radius

A fold does not go through a line. It goes round a small arc, and the arc uses more paper than the stack advances by — a fraction of a millimetre per crease, and several millimetres across a grid, which is why an ambitious tessellation comes out short.

The designer's grid is the dearest thing here

Two hundred and fifty-six panels of box-pleating grid take two hundred and fifty-six search steps to letter — exactly one per panel, at every size from two divisions to sixteen, with not one decision withdrawn. That is the most any pattern in this collection costs per panel of paper. A twist tessellation costs half of it, and a tilted corrugation a quarter.

The other grid

Box pleating is drawn at forty-five degrees, and the twenty-two-and-a-half-degree grid is usually described as the same thing done finer. It is not a refinement, it is a different alphabet: five kinds of vertex become fifty-six, and the share of letterings whose decision needs a search falls from 60 per cent to 22. A finer grid is a larger vocabulary and a less ambiguous one.

The sheet remembers

Perfect memory is the fourth idealisation, and the least examined of the four. It is usually read as a complaint that paper will not lie flat again; the large half is the opposite. A sheet folded once is no longer blank, so folding a second model into it is folding the union of two patterns — and a union folds flat only where every new crease meets every old one at a right angle.

The whole alphabet of a grid

Box pleating is defended as a trade — give up packing efficiency, buy creases that land where they should. There is a third thing it buys and it is much stronger than either: on a forty-five degree grid there are exactly six kinds of interior vertex a flat-foldable design can contain, ever. On a thirty degree grid there are thirty.

What a grid costs in circuits

Box-pleating puts every crease on a square grid, and a square grid is the shape with the most short circuits per panel that this collection draws. On the sixteen-by-sixteen grid a designer actually works on, one mountain-valley labelling in a hundred agrees with itself. A search still finds one in two hundred and sixty-one steps.

What the grid settles

Box pleating is usually defended as a trade: give up efficiency, buy creases that land where they should. There is a second thing it buys and nobody quotes it — on a lattice the best possible packing is a finite question with an answer, while off the lattice nobody knows the best packing of six circles in a square and probably never will.

Every generator · The designing a base field · The patterns a reader can fold